The Wilcoxon Matched-Pairs Signed-Ranks Test

Example:W+ = 5, W- = 40, N = 9, p <= 0

The observation pairs (either a pair each line or the difference itself)Characteristics:
A most usefull test to see whether the members of a pair differ in
size. It resembles the
Sign-Test in scope, but it is much more sensitive. In fact, for
large numbers it is almost as sensitive as the
Student t-test. For small numbers with unknown distributions this test is
even more sensitive than the Student t-test.
As it is only on rare occasions that we do know that values are Normal
distributed, this test is to be preferred over the Student t-test.

H0:
The difference (d = x - y) between the members of each pair (x, y) has
median value zero. To be complete, x an y have identical distributions.

Assumptions:
The distribution of the difference (d) between the values within each pair
(x, y) must be symmetrical, the median difference must be identical to the mean
difference.
As members of a pair are assumed to have identical distributions, their
differences (under H0) should always have a symmetrical
distribution, so this assumption is not very restrictive.

Scale:
Between ordinal and interval (also called an ordered metric scale). It
must be possible to rank the differences.

Procedure:
Rank the differences without regard to the sign of the difference (i.e.,
rank order the absolute differences). Ignore all zero differences
(i.e., pairs with equal members, x=y). Affix the original signs to the rank
numbers. All pairs with equal absolute differences (ties) get the same
rank: all are ranked with the mean of the rank numbers that would have been
assigned if they would have been different.
Sum all positve ranks (W+) and all negative ranks (W-) and
determine the total number of pairs (N).

Level of Significance:
The level of significance is calculated by dividing the number of all
distributions of signs over the ranks that have a SUM(+ranks) <= W+
(if W+ < W-) by 2**N (i.e., the total number of possible
distributions of signs).
These values are tabulated and the level of significance can be looked up.

Remarks:
It is not quite clear why this test is so impopular. It could be due to the fact that
ranking the differences (and calculating the tables) cannot be done with a desk
calculator or be programmed in C easily. When this is not a problem, this test
should realy be preferred over the Student t-test (at least when
N < 50). The Student t-test
is much too vulnerable to deviations from the normal distribution.
Note that the Wilcoxon Matched-Pairs Signed-Ranks Test uses the sizes
of the differences. The result can differ from that of the Sign-test, which
uses the number of + and - signs of the differences.
For N <= 20,
exact probabilities are calculated, for N > 20, the Normal
approximation is used.
A perl script of the test is available
here.
A minimalist Windows version (with dosperl interpreter) is available
here
(<500 kB).